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Theorem ffnafv 28013
Description: A function maps to a class to which all values belong, analogous to ffnfv 5896. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
ffnafv  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem ffnafv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ffn 5593 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fafvelrn 28012 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F''' x )  e.  B
)
32ralrimiva 2791 . . 3  |-  ( F : A --> B  ->  A. x  e.  A  ( F''' x )  e.  B
)
41, 3jca 520 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
) )
5 simpl 445 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  F  Fn  A )
6 afvelrnb0 28006 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  y ) )
7 nfra1 2758 . . . . . 6  |-  F/ x A. x  e.  A  ( F''' x )  e.  B
8 nfv 1630 . . . . . 6  |-  F/ x  y  e.  B
9 rsp 2768 . . . . . . 7  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( F''' x )  e.  B
) )
10 eleq1 2498 . . . . . . . 8  |-  ( ( F''' x )  =  y  ->  ( ( F''' x )  e.  B  <->  y  e.  B ) )
1110biimpcd 217 . . . . . . 7  |-  ( ( F''' x )  e.  B  ->  ( ( F''' x )  =  y  ->  y  e.  B ) )
129, 11syl6 32 . . . . . 6  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( ( F''' x )  =  y  ->  y  e.  B ) ) )
137, 8, 12rexlimd 2829 . . . . 5  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( E. x  e.  A  ( F''' x )  =  y  ->  y  e.  B ) )
146, 13sylan9 640 . . . 4  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ( y  e.  ran  F  ->  y  e.  B ) )
1514ssrdv 3356 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ran  F  C_  B )
16 df-f 5460 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
175, 15, 16sylanbrc 647 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  F : A
--> B )
184, 17impbii 182 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322   ran crn 4881    Fn wfn 5451   -->wf 5452  '''cafv 27950
This theorem is referenced by:  ffnaov  28041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-dfat 27952  df-afv 27953
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